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Optical properties of pyrazine-bridged ruthenium chains by the two-band Hubbard Hamiltonian
1 April 1994
CHEMICAL
PHYSICS Chemical Physics Letters 220 (1994) 327-330
Optical properties of pyrazine-bridged ruthenium chains by the two-band Hubbard Hamiltonian Alessandro Ferretti, Alessandro Lami Istitutodi Chimica Quantisticaed Energetica Molecolarede1CNR, ViaRisorgimento35, 56126 Piss, Italy
Received 19 November 1993;in final form 27 January 1994
Wehave investigated the optical conductivity spectra of [ Ru ( pzRu ) 4]m+ (pz = pyrazine ) , as representative of the whole class of linear mixed valence complexes of ruthenium, through a two-band Hubbard Hamiltonian. It is shown that the main features of absorption spectra, observed experimentally for various degrees of oxidation (m), are well reproduced within this simple model.
1. IIltrodlu!tlon
Mixed-valence compounds,i.e. bi- or poly-metallit complexes in which the metal ions, bridged by ligands, may be present in two or more different oxidation states, are of great interest for several reasons. They are extensively present in living matter as active centers in me&&-enzymes (iron in ferredoxins, manganese in photosynthetic units) and furnish ideal model systems in which electron transfer may take place intramolecularly (see e.g. papers in refs. [ 1,2 ] ) . Furthermore, they are potentially interesting materials for molecular electronics and for superconductivity. In this respect one may notice the analogy with the copper oxide-based high-T, superconductors, which contain doping-activated mixed valence centers, Cu(II)/Cu(III), bridged by oxide ions. A special interest, also for historical reasons, has to be attributed to the Creutz-Taube ion [ ( NH3 )5-Ru-PyRu-W-U15+, orbriefly [ Ru-py-Ru ] ‘+, and to the members of the family that originates from it either by oxidation/reduction, i.e. [ Ru-py-Ru16+/ [ Rupy-Ru ] ‘+ [ 3 1, or by prolongation of the chain, [ Ru-
(PY-RuLl”+, where the total charge n varies depending on the oxidation state [ 4 1. From the beginning of this research area, the problem of localization/delocalization of electrons captured much attention. The problem was to understand if the two ruthenium ions in the Creutz-Taube complex were equivalent (i.e. both in the intermediate oxidation state and with a charge 2.5 ) or inequivalent (i.e. one ruthenium (III) and one ruthenium (II)). A large amount of experimental investigations now supports the equivalence of the two ions [ 5 1. In any case this initial question, closely related to the possibility of distortion induced by the pseudoJahn-Teller effect, addressed much theoretical work on the role of vibronic interactions in determining the features of absorption spectra [6-91 as well as of other observables, like the EPR g-tensor, for those compounds which have unpaired electrons [ 6, IO,1 11. In the present Letter, for the first time in mixed valence chains, we will focus on a different aspect, i.e. the possible role of correlation in determining some peculiar features of the absorption spectra, and their dependence on the total number of electrons. In par-
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights mserved SSDZOOO9-2614(94) 00157-L
A. Ferretti, A. L.ami /Chemical PhysicsLetters 220 (1994) 327-330
328
ticular we will show that a simple Hubbard Hamiltonian may reproduce the characteristic shift of spectral weight to a low-frequency band, as observed in a chain of pyrazine-bridged ruthenium atoms, upon progressive oxidation [ 4 1.
x(z)
jp =it
2. The model As far as what concerns properties related to backbonding, linear mixed valence complexes, such as the Creutz-Taube analogues of ref. [ 41, may be simply modelled as an alternating ML chain with one orbital per site, e.g. the d orbital with the right symmetry for the metal M and the R* orbital for the ligand L [ 12,131. Within this picture, the minimal model Hamiltonian that one may conceive for describing electronic structure and related properties for this compounds, is the well known two-band Hubbard model. For electrons in a chain of alternating M and L, the Hubbard Hamiltonian may be written as HZ C Ejnj,b+t C (Ujt,Uj+i,,+h.C.)
j.0
i.s
j
where a,$ ( qn,,) is the creation (annihilation) operator for one electron in the orbital of sitej with spin a, nj,s=a,$aj,O, ej=eL, cM is the site energy and the parameters t and U are respectively the hopping of one electron between orbitals on adjacent M and L sites and the Coulomb repulsion for two electrons on the same M orbital. In this Letter, we have studied the optical properties of the Hamiltonian of Eq. ( 1 ), showing that the essential features observed experimentally [ 41 at low frequency may be reproduced within this simple picture. As usual with the Hubbard Hamiltonian [ 1417 1, we have calculated optical conductivity, which gives a representation of photon absorption. The optical conductivity (a’(o) ) is the real part of the linear-response conductivity of a system in a uniform time-dependent electric field (which in turn is proportional to the imaginary part of the suscettivity ,y) . We report here the relevant formulae for finite systems such as the M( LM), chains under study, while a detailed derivation may be found in ref. [ 17 1,
d ImX(w+iq)
a’(w)=-
=
,
+w&+&-fo-!w)
C j.0
(~,t,aj+l,u-aj+,l,&j,~)
2
,
(2)
where N is the total number of sites (N= Nr,,+ NL), is the current-current correlation function, J& is the ground state energy, H the Hamiltonian and_& is the ‘paramagnetic’ current induced by the electric field. The Lanczos technique has been utilized here to calculate a’(w) since it is particularly suitable for obtaining the coefficients (Y,and /In directly for the continued fraction expansion of x( z) . The convergence for the essential optical features is obtained typically after 100 Lanczos steps. A value of ~=0.2 has been chosen to obtain a Lorentzian shape of the peaks.
x(z)
3. Results In a previous paper [ 13 ] the optical properties of the Hamiltonian of Eq. (1) were investigated in a general manner, in order to find out the possible usefulness of the model for studying mixed valence complexes. We found that, for some values of the parameters, a good reproduction of the experimental spectra for the system [Ru(pzRu)]“+ (n=4, 5, 6) was achieved, especially for the low-frequency behaviour. These results have encouraged further investigation on longer chain analogues of these ions (e.g. M (LM ) ,, with n > 1) , since experimental spectra are available from the literature [ 41. In this Letter, we report some preliminary results, while in a future publication a more extensive study of the Hamiltonian of Eq. ( 1) for modelling mixed valence complexes of Ru and pyrazine will be reported. In our approach, Ru (pzRu), have one orbital per site (d, for Ru and E* for pyrazine, for the pz ring in the xy plane) and the vacuum (all orbitals empty) has Ru atoms in the form of Ru4+. The systems with one electron per Ru atom correspond then to the fully oxidized species Ru ( pzRu ) !,+ ’ j3+ and each further addition of one electron gives rise to the reduction of
:.: : I
329
A. Ferretti, A. Lami /Chemical Physics Letters 220 (1994) 327-330
a Ru atom from + 3 to + 2, until the full reduced species Ru(pzRu)~+‘)*+ is obtained. Absorption spectra for various Ru(pzRu), systems with different n and degrees of oxidation, were measured by von Kameke, Tom and Taube [ 4 1. They found that in the region between 800 and 200 nm ( 52:1S-6.2 eV) three main bands appeared, a broad one around 800-500 nm (a 1.5-2.5 eV) and two around 300-200 nm ( =4.2-6.2 eV). Furthermore, in the near-infrared (below 1.5 eV) there was no absorption for both fully oxidized and fully reduced species, but it was observed for intermediate degrees of oxidation. In that paper the authors tried to assign the low-frequency absorption bands as end to end intervalence transitions, where, in a localized picture one has two states rather close in energy with different site population distribution. This picture is however a simple one. It is well known that in CT complexes the strong backbond interaction causes a strong delocalization [ 51 for the electrons added to the Ru(pzR~)p+‘)~+ species and this does not fit with the above assignments. Our approach considers delocalized electrons on the lattice (the term t in Eq. ( 1) ) , with the inclusion of the maiu source of correlation, the Coulomb repulsion (the term U in Eq. ( 1) ) and then contains all the essential (and minimal) ingredients for modelling CT complexes. We will see that with this model all the essential spectral features observed experimentally may be reproduced and interpreted. It is quite obvious, in order to have a quantitative agreement, further ingredients should be added to the present minimal model. From the experimental bands at high frequency, one may suppose that the values of A (charge transfer energy; A=e,_- cM) and U to be considered in the computations are around 5-6 eV. Furthermore, a hopping t= -0.5 eV can be realistic for describing systems with a certain amount of delocalization. With these parameters (precisely U= 5, eL= - 1 and EM= - 7, e.g. A= 6; all in eV) the optical conductivity spectrum of Ru( pzRu)Y+ (e.g. the system of Fig. 2 of ref. [ 41) has been computed at various degrees of oxidation in the range 10 < m < 15 and it is reported inFig.
1.
For the simplicity of the model the results are impressive. Above 3.5 eV, where all spectra intersect and invert their order in spectral weight, there are two
0.6
g 0
: : : .:
: : : :
-
m=15
---
m=14
---
m=13
-_-
m42
-.-
m=ll
______m=lO
Fig. 1. Optical conductivityspectrafor Ru(pzRu)T+ eV, U= 5 eV and A= 6 eV) at different M.
(t= -0.5
bands respectively at x 4-5 eV and = 5.5-7 eV which, except for their intensity, correspond to the bands at high energy observed experimentally. The spectral weight diminishes as the degree of oxidation increases, in agreement with experiments, the bands being due to CT transitions from a single occupied metal to the l&and, whose amount decreases as m grows (growing the number of electrons i.e. double occupation). At low energy, below 2.5 eV, there is a broad band whose intensity increases and the maximum moves toward lower energies as m grows, still in good agreement with experiments. This band corresponds to CT transitions from double occupied metal sites to the l&and. At even lower energies both the full oxidized and the full reduced species do not show absorption, while a peak around 0.2 eV characterizes the spectra of the species with intermediate degrees of oxidation. The shoulder observed for m = 10 is fictitious: q is too large, even if it gives a good graphic representation of the bands and in order to avoid divergence at w z 0 we have replaced l/o with o/(w*+$) in the computations. The peak around 0.2 eV is expected to drop to w=O as the lattice length grows being the Drude precursor discussed in previous papers [ 13,171. It corre-
330
A. Ferretti,A. Lomi /Chemical PhysicsLetters 220 (I 994) 327-330
spends, in the infinite chain limit, to the free electron contribution to the conductivity. A global overlook to Fig. 1 reveals that there is, as in the experiments, a transfer of spectral weight from, high to low energy. This had already been observed in Cu-0 high-T, materials [ 18,191, where the Hubbard model has been widely studied in the perspective of understanding the mechanism of superconductivity. As a concluding remark, from the results obtained the two-band Hubbard Hamiltonian appears to be capable of capturing the essential features of the op tical properties of bridged mixed valence complexes of ruthenium. Other ingredients, such as vibronic coupling or the inclusion of other orbitals, may further improve the model and give a better understanding of mixed valence compounds. In particular, it has to be noticed that our model is expected to be more sensitive to vibronic coupling in the limit of weak delocalization (i.e. small hopping t ) . Acknowledgement The authors are grateful to Professor Mary Jo Ondrechen for helpful suggestions and stimulating discussions. This research has been performed in the framework of ‘Progetto Finalizzato Materiali Speciali per Tecnologie Avanzate de1 Consiglio Nazionale delle Ricerche’. References [l]M.K. Johnson, R.B.King,D.M. Kurtz Jr., C. Kutal, M.L. Norton and RA. Scott, eds., Advan. Chem. Series 226, Electron transfer in biology and solid state (American Chemical Society, Washington, 1990).
[ 21 J.R. Bolton, N. Mataga and G. McLendon, eds., Advan. Chem. Series 228, Electron transfer in inorganic, organic and biological systems (American Chemical Society, Washington, 1991). [3] C. CreutzandH. Taube, J.Am. Chem. Sot. 95 (1973) 1086. [4] A. von Kameke, G.M. Tom and H. Taube, Inorg. Chcm. 17 (1978) 1790. [ 5] LB. Bersuker and S.A. Borshch, in: Advan. Chem. Phys. Vol. 81, eds.I. Prigogine and S.A. Rice (Wiley, New York, 1992) p. 703. [6] K. Neuenschwander, S.B. Piepho and P.N. Schatz, J. Am. Chem. Sot. 107 (1985) 7862. [ 7 ] K. Prassides, P.N. Shatz, K.J. Wong and P. Day, J. Phys. Chem. 90 (1986) 5588. [ 81 J. Ko, L.-T. Zhang and M.J. Ondrechen, J. Am. Chem. Sot. 109 (1987) 1672. [9] K Prassides and P.N. Shatz, Chem. Phys. Letters 178 (1991) 227. [ lo] L.-T. Zhang, J. Ko and M.J. Gndrechen, J. Am. Chem. Sot. 109 (1987) 1667. [ 111 J. Ko, L.-T. Zhang and M.J. Ondrcchen, J. Am. Chem. Sot. 109 (1987) 1672. [ 121 M.J. Gndrechen, S. Gozashti and X.M. Wu, J. Chem. Phys. 96 (1992) 3255. [ 131 A. Ferretti and A. Lami, Chem. Phys., in press. [ 141 W. Kohn, Phys. Rev. 133 (1964) A171. [ 151 P.F. Maldague, Phys. Rev. B 16 (1977) 2437. [ 161 B.S. Shastry and B. Sutherland, Phys. Rev. Letters 65 (1990) 243. [ 17 ] J. Wagner, W. Hanke and DJ. Scalapino, Phys. Rev. B 43 (1991) 10517. [ 18 ] S.L. Cooper, G.A. Thomas, J. Grenstein, D.H. Rapkine, M. Capizzi, T. Timusk, A.J. Millis, L.F. Schneemeyer and J.W. Waszczak, Phys. Rev. B 40 (1989) 11358. [ 191 S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura and S. Tajima, Phys. Rev. B 43 (1991) 7942.
CHEMICAL
PHYSICS Chemical Physics Letters 220 (1994) 327-330
Optical properties of pyrazine-bridged ruthenium chains by the two-band Hubbard Hamiltonian Alessandro Ferretti, Alessandro Lami Istitutodi Chimica Quantisticaed Energetica Molecolarede1CNR, ViaRisorgimento35, 56126 Piss, Italy
Received 19 November 1993;in final form 27 January 1994
Wehave investigated the optical conductivity spectra of [ Ru ( pzRu ) 4]m+ (pz = pyrazine ) , as representative of the whole class of linear mixed valence complexes of ruthenium, through a two-band Hubbard Hamiltonian. It is shown that the main features of absorption spectra, observed experimentally for various degrees of oxidation (m), are well reproduced within this simple model.
1. IIltrodlu!tlon
Mixed-valence compounds,i.e. bi- or poly-metallit complexes in which the metal ions, bridged by ligands, may be present in two or more different oxidation states, are of great interest for several reasons. They are extensively present in living matter as active centers in me&&-enzymes (iron in ferredoxins, manganese in photosynthetic units) and furnish ideal model systems in which electron transfer may take place intramolecularly (see e.g. papers in refs. [ 1,2 ] ) . Furthermore, they are potentially interesting materials for molecular electronics and for superconductivity. In this respect one may notice the analogy with the copper oxide-based high-T, superconductors, which contain doping-activated mixed valence centers, Cu(II)/Cu(III), bridged by oxide ions. A special interest, also for historical reasons, has to be attributed to the Creutz-Taube ion [ ( NH3 )5-Ru-PyRu-W-U15+, orbriefly [ Ru-py-Ru ] ‘+, and to the members of the family that originates from it either by oxidation/reduction, i.e. [ Ru-py-Ru16+/ [ Rupy-Ru ] ‘+ [ 3 1, or by prolongation of the chain, [ Ru-
(PY-RuLl”+, where the total charge n varies depending on the oxidation state [ 4 1. From the beginning of this research area, the problem of localization/delocalization of electrons captured much attention. The problem was to understand if the two ruthenium ions in the Creutz-Taube complex were equivalent (i.e. both in the intermediate oxidation state and with a charge 2.5 ) or inequivalent (i.e. one ruthenium (III) and one ruthenium (II)). A large amount of experimental investigations now supports the equivalence of the two ions [ 5 1. In any case this initial question, closely related to the possibility of distortion induced by the pseudoJahn-Teller effect, addressed much theoretical work on the role of vibronic interactions in determining the features of absorption spectra [6-91 as well as of other observables, like the EPR g-tensor, for those compounds which have unpaired electrons [ 6, IO,1 11. In the present Letter, for the first time in mixed valence chains, we will focus on a different aspect, i.e. the possible role of correlation in determining some peculiar features of the absorption spectra, and their dependence on the total number of electrons. In par-
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights mserved SSDZOOO9-2614(94) 00157-L
A. Ferretti, A. L.ami /Chemical PhysicsLetters 220 (1994) 327-330
328
ticular we will show that a simple Hubbard Hamiltonian may reproduce the characteristic shift of spectral weight to a low-frequency band, as observed in a chain of pyrazine-bridged ruthenium atoms, upon progressive oxidation [ 4 1.
x(z)
jp =it
2. The model As far as what concerns properties related to backbonding, linear mixed valence complexes, such as the Creutz-Taube analogues of ref. [ 41, may be simply modelled as an alternating ML chain with one orbital per site, e.g. the d orbital with the right symmetry for the metal M and the R* orbital for the ligand L [ 12,131. Within this picture, the minimal model Hamiltonian that one may conceive for describing electronic structure and related properties for this compounds, is the well known two-band Hubbard model. For electrons in a chain of alternating M and L, the Hubbard Hamiltonian may be written as HZ C Ejnj,b+t C (Ujt,Uj+i,,+h.C.)
j.0
i.s
j
where a,$ ( qn,,) is the creation (annihilation) operator for one electron in the orbital of sitej with spin a, nj,s=a,$aj,O, ej=eL, cM is the site energy and the parameters t and U are respectively the hopping of one electron between orbitals on adjacent M and L sites and the Coulomb repulsion for two electrons on the same M orbital. In this Letter, we have studied the optical properties of the Hamiltonian of Eq. ( 1 ), showing that the essential features observed experimentally [ 41 at low frequency may be reproduced within this simple picture. As usual with the Hubbard Hamiltonian [ 1417 1, we have calculated optical conductivity, which gives a representation of photon absorption. The optical conductivity (a’(o) ) is the real part of the linear-response conductivity of a system in a uniform time-dependent electric field (which in turn is proportional to the imaginary part of the suscettivity ,y) . We report here the relevant formulae for finite systems such as the M( LM), chains under study, while a detailed derivation may be found in ref. [ 17 1,
d ImX(w+iq)
a’(w)=-
=
,
+w&+&-fo-!w)
C j.0
(~,t,aj+l,u-aj+,l,&j,~)
2
,
(2)
where N is the total number of sites (N= Nr,,+ NL), is the current-current correlation function, J& is the ground state energy, H the Hamiltonian and_& is the ‘paramagnetic’ current induced by the electric field. The Lanczos technique has been utilized here to calculate a’(w) since it is particularly suitable for obtaining the coefficients (Y,and /In directly for the continued fraction expansion of x( z) . The convergence for the essential optical features is obtained typically after 100 Lanczos steps. A value of ~=0.2 has been chosen to obtain a Lorentzian shape of the peaks.
x(z)
3. Results In a previous paper [ 13 ] the optical properties of the Hamiltonian of Eq. (1) were investigated in a general manner, in order to find out the possible usefulness of the model for studying mixed valence complexes. We found that, for some values of the parameters, a good reproduction of the experimental spectra for the system [Ru(pzRu)]“+ (n=4, 5, 6) was achieved, especially for the low-frequency behaviour. These results have encouraged further investigation on longer chain analogues of these ions (e.g. M (LM ) ,, with n > 1) , since experimental spectra are available from the literature [ 41. In this Letter, we report some preliminary results, while in a future publication a more extensive study of the Hamiltonian of Eq. ( 1) for modelling mixed valence complexes of Ru and pyrazine will be reported. In our approach, Ru (pzRu), have one orbital per site (d, for Ru and E* for pyrazine, for the pz ring in the xy plane) and the vacuum (all orbitals empty) has Ru atoms in the form of Ru4+. The systems with one electron per Ru atom correspond then to the fully oxidized species Ru ( pzRu ) !,+ ’ j3+ and each further addition of one electron gives rise to the reduction of
:.: : I
329
A. Ferretti, A. Lami /Chemical Physics Letters 220 (1994) 327-330
a Ru atom from + 3 to + 2, until the full reduced species Ru(pzRu)~+‘)*+ is obtained. Absorption spectra for various Ru(pzRu), systems with different n and degrees of oxidation, were measured by von Kameke, Tom and Taube [ 4 1. They found that in the region between 800 and 200 nm ( 52:1S-6.2 eV) three main bands appeared, a broad one around 800-500 nm (a 1.5-2.5 eV) and two around 300-200 nm ( =4.2-6.2 eV). Furthermore, in the near-infrared (below 1.5 eV) there was no absorption for both fully oxidized and fully reduced species, but it was observed for intermediate degrees of oxidation. In that paper the authors tried to assign the low-frequency absorption bands as end to end intervalence transitions, where, in a localized picture one has two states rather close in energy with different site population distribution. This picture is however a simple one. It is well known that in CT complexes the strong backbond interaction causes a strong delocalization [ 51 for the electrons added to the Ru(pzR~)p+‘)~+ species and this does not fit with the above assignments. Our approach considers delocalized electrons on the lattice (the term t in Eq. ( 1) ) , with the inclusion of the maiu source of correlation, the Coulomb repulsion (the term U in Eq. ( 1) ) and then contains all the essential (and minimal) ingredients for modelling CT complexes. We will see that with this model all the essential spectral features observed experimentally may be reproduced and interpreted. It is quite obvious, in order to have a quantitative agreement, further ingredients should be added to the present minimal model. From the experimental bands at high frequency, one may suppose that the values of A (charge transfer energy; A=e,_- cM) and U to be considered in the computations are around 5-6 eV. Furthermore, a hopping t= -0.5 eV can be realistic for describing systems with a certain amount of delocalization. With these parameters (precisely U= 5, eL= - 1 and EM= - 7, e.g. A= 6; all in eV) the optical conductivity spectrum of Ru( pzRu)Y+ (e.g. the system of Fig. 2 of ref. [ 41) has been computed at various degrees of oxidation in the range 10 < m < 15 and it is reported inFig.
1.
For the simplicity of the model the results are impressive. Above 3.5 eV, where all spectra intersect and invert their order in spectral weight, there are two
0.6
g 0
: : : .:
: : : :
-
m=15
---
m=14
---
m=13
-_-
m42
-.-
m=ll
______m=lO
Fig. 1. Optical conductivityspectrafor Ru(pzRu)T+ eV, U= 5 eV and A= 6 eV) at different M.
(t= -0.5
bands respectively at x 4-5 eV and = 5.5-7 eV which, except for their intensity, correspond to the bands at high energy observed experimentally. The spectral weight diminishes as the degree of oxidation increases, in agreement with experiments, the bands being due to CT transitions from a single occupied metal to the l&and, whose amount decreases as m grows (growing the number of electrons i.e. double occupation). At low energy, below 2.5 eV, there is a broad band whose intensity increases and the maximum moves toward lower energies as m grows, still in good agreement with experiments. This band corresponds to CT transitions from double occupied metal sites to the l&and. At even lower energies both the full oxidized and the full reduced species do not show absorption, while a peak around 0.2 eV characterizes the spectra of the species with intermediate degrees of oxidation. The shoulder observed for m = 10 is fictitious: q is too large, even if it gives a good graphic representation of the bands and in order to avoid divergence at w z 0 we have replaced l/o with o/(w*+$) in the computations. The peak around 0.2 eV is expected to drop to w=O as the lattice length grows being the Drude precursor discussed in previous papers [ 13,171. It corre-
330
A. Ferretti,A. Lomi /Chemical PhysicsLetters 220 (I 994) 327-330
spends, in the infinite chain limit, to the free electron contribution to the conductivity. A global overlook to Fig. 1 reveals that there is, as in the experiments, a transfer of spectral weight from, high to low energy. This had already been observed in Cu-0 high-T, materials [ 18,191, where the Hubbard model has been widely studied in the perspective of understanding the mechanism of superconductivity. As a concluding remark, from the results obtained the two-band Hubbard Hamiltonian appears to be capable of capturing the essential features of the op tical properties of bridged mixed valence complexes of ruthenium. Other ingredients, such as vibronic coupling or the inclusion of other orbitals, may further improve the model and give a better understanding of mixed valence compounds. In particular, it has to be noticed that our model is expected to be more sensitive to vibronic coupling in the limit of weak delocalization (i.e. small hopping t ) . Acknowledgement The authors are grateful to Professor Mary Jo Ondrechen for helpful suggestions and stimulating discussions. This research has been performed in the framework of ‘Progetto Finalizzato Materiali Speciali per Tecnologie Avanzate de1 Consiglio Nazionale delle Ricerche’. References [l]M.K. Johnson, R.B.King,D.M. Kurtz Jr., C. Kutal, M.L. Norton and RA. Scott, eds., Advan. Chem. Series 226, Electron transfer in biology and solid state (American Chemical Society, Washington, 1990).
[ 21 J.R. Bolton, N. Mataga and G. McLendon, eds., Advan. Chem. Series 228, Electron transfer in inorganic, organic and biological systems (American Chemical Society, Washington, 1991). [3] C. CreutzandH. Taube, J.Am. Chem. Sot. 95 (1973) 1086. [4] A. von Kameke, G.M. Tom and H. Taube, Inorg. Chcm. 17 (1978) 1790. [ 5] LB. Bersuker and S.A. Borshch, in: Advan. Chem. Phys. Vol. 81, eds.I. Prigogine and S.A. Rice (Wiley, New York, 1992) p. 703. [6] K. Neuenschwander, S.B. Piepho and P.N. Schatz, J. Am. Chem. Sot. 107 (1985) 7862. [ 7 ] K. Prassides, P.N. Shatz, K.J. Wong and P. Day, J. Phys. Chem. 90 (1986) 5588. [ 81 J. Ko, L.-T. Zhang and M.J. Ondrechen, J. Am. Chem. Sot. 109 (1987) 1672. [9] K Prassides and P.N. Shatz, Chem. Phys. Letters 178 (1991) 227. [ lo] L.-T. Zhang, J. Ko and M.J. Gndrechen, J. Am. Chem. Sot. 109 (1987) 1667. [ 111 J. Ko, L.-T. Zhang and M.J. Ondrcchen, J. Am. Chem. Sot. 109 (1987) 1672. [ 121 M.J. Gndrechen, S. Gozashti and X.M. Wu, J. Chem. Phys. 96 (1992) 3255. [ 131 A. Ferretti and A. Lami, Chem. Phys., in press. [ 141 W. Kohn, Phys. Rev. 133 (1964) A171. [ 151 P.F. Maldague, Phys. Rev. B 16 (1977) 2437. [ 161 B.S. Shastry and B. Sutherland, Phys. Rev. Letters 65 (1990) 243. [ 17 ] J. Wagner, W. Hanke and DJ. Scalapino, Phys. Rev. B 43 (1991) 10517. [ 18 ] S.L. Cooper, G.A. Thomas, J. Grenstein, D.H. Rapkine, M. Capizzi, T. Timusk, A.J. Millis, L.F. Schneemeyer and J.W. Waszczak, Phys. Rev. B 40 (1989) 11358. [ 191 S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura and S. Tajima, Phys. Rev. B 43 (1991) 7942.